Yes, the binomial expansion can be extended to handle negative and fractional powers, though the process is more complex than for positive integer powers. For these cases, the binomial series becomes an infinite series. The general formula involves the concept of factorials and the binomial coefficients are calculated using a generalised formula. However, it's essential to note that the convergence of the series depends on the values of a and b. For IB Mathematics Applications & Interpretations, the primary focus is on positive integer powers, but it's good to be aware that the concept can be extended further in advanced mathematics.

Yes, while Pascal's triangle is a powerful tool for quickly determining the coefficients in a binomial expansion, it has its limitations. The primary limitation is its practicality for large values of n. As you move down the triangle, the numbers grow rapidly, making it cumbersome to manually construct and read for higher powers. For small values of n, Pascal's triangle is efficient, but for larger values, using the combination formula or other methods to compute binomial coefficients directly becomes more practical. Additionally, Pascal's triangle only applies to binomial expansions and not to expansions with more than two terms.

The binomial expansion has a direct application in probability theory, especially in problems involving binomial experiments. In a binomial experiment, there are only two possible outcomes (like success and failure), and the experiment is repeated a fixed number of times. The probability of getting a specific number of successes can be determined using the coefficients from the binomial expansion. The binomial coefficient "n choose k" represents the number of ways to achieve k successes in n trials. Multiplying this by the probability of success raised to the power k and the probability of failure raised to the power (n-k) gives the probability of that specific outcome. This connection between algebra and probability is a beautiful illustration of how different mathematical concepts can come together to solve real-world problems.

The binomial theorem and combinatorics are intrinsically linked through the concept of binomial coefficients. In the binomial expansion, the coefficients represent the number of ways to choose a certain number of items from a larger set, without regard to order. This is a fundamental idea in combinatorics known as combinations. For instance, the coefficient of a^k * b^(n-k) in the expansion of (a+b)ⁿ is "n choose k", which denotes the number of ways to select k items from a set of n. Thus, the binomial theorem provides a bridge between algebraic expansions and combinatorial counting, showcasing the interconnectedness of different areas of mathematics.

The binomial expansion is a foundational concept in algebra that offers a systematic way to expand expressions of the form (a+b)ⁿ, where n is a positive integer. Its importance stems from its wide applicability in various mathematical problems, including polynomial approximations, probability theory, and combinatorics. By understanding the binomial expansion, students can tackle a range of problems that would be cumbersome or nearly impossible to solve without this tool. Moreover, the principles behind the binomial expansion pave the way for more advanced topics in mathematics, ensuring that students have a solid grounding as they progress in their studies.